×

zbMATH — the first resource for mathematics

On fractional integration formulae for Aleph functions. (English) Zbl 1242.33021
Summary: This paper is devoted to the study of a new special function, which is called Aleph function, according to the symbol used to represent this function. This function is an extension of the \(I\)-function, which itself is a generalization of the well-known and familiar \(G\)- and \(H\)-functions in one variable. In this paper, a notation and a complete definition of the Aleph function will be presented. Fractional integration of Aleph functions, in which the argument of the Aleph function contains a factor \(t^{\lambda }(1 - t)^{\mu }, \lambda , \mu > 0\), will be investigated. The results derived are of most general character and include many results given earlier by various authors including A. A. Kilbas [Fract. Calc. Appl. Anal. 8, No. 2, 113–126 (2005; Zbl 1144.26008)], A. A. Kilbas and M. Saigo [Fukuoka Univ. Sci. Rep. 28, No. 2, 41–51 (1998; Zbl 0923.33006)] and L. Galué [Int. J. Appl. Math. 10, No. 3, 255–267 (2002; Zbl 1036.33002)] and others. The results obtained form the key formulae for the results on various potentially useful special functions of physical and biological sciences and technology available in the literature.

MSC:
33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)
26A33 Fractional derivatives and integrals
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Buschman, R.G.; Srivastava, H.M., The \(\overline{H}\)-function associated with certain class of Feynman integrals, J. phys. A.: math. gen., 23, 4707-4710, (1990) · Zbl 0695.33002
[2] Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F.G., Higher transcendental functions, Vol. 1, (1953), McGraw-Hill New York · Zbl 0052.29502
[3] Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F.G., Higher transcendental functions, Vol. 2, (1954), McGraw-Hill New York · Zbl 0051.30303
[4] Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F.G., Higher transcendental functions, Vol. 3, (1955), McGraw-Hill New York · Zbl 0064.06302
[5] Fox, C., The G and H-functions as symmetrical Fourier kernels, Trans amer. math. soc., 98, 385-429, (1961) · Zbl 0096.30804
[6] Galué, L., Differintegrals of wright’s generalized hypergeometric functions, Int. J. appl. math., 10, 255-267, (2002) · Zbl 1036.33002
[7] ()
[8] Inayat-Hussain, A.A., New properties of generalized hypergeometric series derivable from Feynman integrals. I. transformation and reduction formulae, J. phys. A, 20, 13, 4109-4117, (1987) · Zbl 0634.33005
[9] Inayat-Hussain, A.A., New properties of generalized hypergeometric series derivable from Feynman integrals. II. A generalisation of the H function, J. phys. A, 20, 13, 4119-4128, (1987) · Zbl 0634.33006
[10] Kilbas, A.A., Fractional calculus of generalized wright function, Frac. calc. appl. anal., 8, 113-126, (2005) · Zbl 1144.26008
[11] Kilbas, A.A.; Saigo, M., Fractional calculus of the H-function, Fukuoka univ. sci. rep., 28, 41-51, (1998) · Zbl 0923.33006
[12] Kilbas, A.A.; Saigo, M., H-transforms: theory and application, (2004), Chapman & Hall/CRC Press Boca Raton, London, New York · Zbl 1056.44001
[13] Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J., Theory and applications of fractional differential equations, North holland · mathematics studies, Vol. 204, (2006), Elsevier New York · Zbl 1092.45003
[14] Kiryakova, V., The special functions of fractional calculus as generalized fractional calculus operators of some basic functions, Comput. math. appl., 59, 3, 1128-1141, (2010) · Zbl 1189.26007
[15] Kiryakova, V., The multiindex mittag – leffler functions as an important class of special functions of fractional calculus, Comput. math. appl., 59, 5, 1885-1895, (2010) · Zbl 1189.33034
[16] Mainardi, F.; Pagnini, G., Salvatore pincherle the pioneer of mellin – barnes integrals, J. comput. appl. math., 153, 331-342, (2003) · Zbl 1050.33018
[17] A.M. Mathai (Ed.), Special Functions and Functions of Matrix Argument: Recent Developments and Recent Applications in Statistics and Astrophysics, Lecture Notes Publication No. 32, Center for Mathematical Sciences, Pala Campus, Pala, India, 2005.
[18] Mathai, A.M.; Saxena, R.K.; Haubold, H.J., The H-function: theory and applications, (2010), Springer New York · Zbl 1223.85008
[19] Pincherle, S., Sulle funzioni ipergeometriche generalizzate, Rom. acc. L. rend., 4IV_{1}, 694-700, (1888), 792-799 · JFM 20.0432.01
[20] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego · Zbl 0918.34010
[21] Prudnikov, A.P.; Brychkov, Yu.A.; Marichev, O.I., Integrals and series, More special functions, Vol. 3, (1990), Gordon and Breach Science Publishers New York · Zbl 0967.00503
[22] Saxena, R.K.; Kumar, R., A basic analogue of generalized H-function, Matematiche (Catania), 50, 263-271, (1995) · Zbl 0899.33012
[23] Saxena, R.K.; Mathai, A.M.; Haubold, H.J., Unified fractional kinetic equations and a diffusion equation, Astrophys. space sci., 290, 299-310, (2004)
[24] Saxena, V.P., Formal solution of certain new pair of dual integral equations involving H-functions, Proc. nat. acad. sci. India sect. A, 52, 366-375, (1982) · Zbl 0535.45001
[25] Srivastava, H.M.; Saxena, R.K., Operators of fractional integration and their applications, Appl. math. comput., 118, 1-52, (2001) · Zbl 1022.26012
[26] Srivastava, H.M.; Gupta, K.C.; Goyal, S.P., The H-functions of one and two variables with applications, (1982), South Asian Publishers New Delhi · Zbl 0506.33007
[27] Srivastava, H.M.; Lin, Shy-Der; Wang, Pin-Yu, Some fractional-calculus results for the \(\overline{H}\)-function associated with a class of Feynman integrals, Russ. J. math. phys., 13, 1, 94-100, (2006) · Zbl 1223.33027
[28] Südland, N.; Baumann, B.; Nonnenmacher, T.F., Who knows about the aleph (\(\aleph\))-functions?, Fract. calc. appl. anal., 1, 4, 401-402, (1998)
[29] Südland, N.; Baumann, B.; Nonnenmacher, T.F., Fractional driftless fokker – planck equation with power law diffusion coefficients, (), 513-525 · Zbl 1044.82011
[30] Wright, E.M., The asymtotic expansion of generalized hypergeometric function, J. London math. soc., 10, 287-293, (1935)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.